Absolute stability and dynamical stabilisation in predator-prey systems

Dagbovie, Ayawoa S.; Sherratt, Jonathan A.

doi:10.1007/s00285-013-0672-8

Cited by 18 publications

(15 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Sect. 6, we have shown how the width of the region where the steady state remains stable can be calculated quantitatively (see also Dagbovie and Sherratt 2013). This may be useful information for the design of experiments, ecological monitoring as well as environmental assessment.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…1 showing that there can be a plateau region behind the invasion front in which the (unstable) co-existence steady state is "dynamically stabilised". Denoting the invasion velocity by V front , the condition for such a plateau is V front > V * , so that the invasion can outrun all growing linear modes (Dagbovie and Sherratt 2013). But the curve λ * (V ) can also form the basis of calculations.…”

Section: Quantitative Calculations Using Absolute Stabilitymentioning

confidence: 99%

“…1. The method used to measure the width in numerical simulations is described in Dagbovie and Sherratt (2013). The parameter values were a = 1.…”

Section: Quantitative Calculations Using Absolute Stabilitymentioning

confidence: 99%

See 2 more Smart Citations

A Mathematical Biologist’s Guide to Absolute and Convective Instability

2013

Self Cite

View full text Add to dashboard Cite

Mathematical models have been highly successful at reproducing the complex spatiotemporal phenomena seen in many biological systems. However, the ability to numerically simulate such phenomena currently far outstrips detailed mathematical understanding. This paper reviews the theory of absolute and convective instability, which has the potential to redress this inbalance in some cases. In spatiotemporal systems, unstable steady states subdivide into two categories. Those that are absolutely unstable are not relevant in applications except as generators of spatial or spatiotemporal patterns, but convectively unstable steady states can occur as persistent features of solutions. The authors explain the concepts of absolute and convective instability, and also the related concepts of remnant and transient instability.They give examples of their use in explaining qualitative transitions in solution behaviour. They then describe how to distinguish different types of instability, focussing on the relatively new approach of the absolute spectrum. They also discuss the use of the theory for making quantitative predictions on how spatiotemporal solutions change with model parameters. The discussion is illustrated throughout by numerical simulations of a model for river-based predator-prey systems.

show abstract

Section: Summary and Discussionmentioning

confidence: 99%

Section: Quantitative Calculations Using Absolute Stabilitymentioning

confidence: 99%

See 1 more Smart Citation

A Mathematical Biologist’s Guide to Absolute and Convective Instability

2013

Self Cite

View full text Add to dashboard Cite

show abstract

“…Even though the phenomenon of dynamical stabilization is difficult to study in reaction-diffusion equations (since it requires at least two coupled equations) much progress has been made in recent years, see e.g. [4,34] and references therein. A scalar reaction-diffusion equation, however, has monotone solution dynamics, and can therefore never support these complex patterns.…”

mentioning

confidence: 99%

Dynamical stabilization and traveling waves in integrodifference equations

Bourgeois¹,

LeBlanc²,

Lutscher³

2020

Discrete &Amp; Continuous Dynamical Systems - S

View full text Add to dashboard Cite

Integrodifference equations are discrete-time analogues of reactiondiffusion equations and can be used to model the spatial spread and invasion of non-native species. They support solutions in the form of traveling waves, and the speed of these waves gives important insights about the speed of biological invasions. Typically, a traveling wave leaves in its wake a stable state of the system. Dynamical stabilization is the phenomenon that an unstable state arises in the wake of such a wave and appears stable for potentially long periods of time, before it is replaced with a stable state via another transition wave. While dynamical stabilization has been studied in systems of reaction-diffusion equations, we here present the first such study for integrodifference equations. We use linear stability analysis of traveling-wave profiles to determine necessary conditions for the emergence of dynamical stabilization and relate it to the theory of stacked fronts. We find that the phenomenon is the norm rather than the exception when the non-spatial dynamics exhibit a stable two-cycle.

show abstract

“…Comparing the stability properties of these two different classes of fronts is a work in progress. We also mention that transition fronts in the Rosenzweig-MacArthur system have been observed numerically and their stability properties have been investigated in [10], but in parameter regimes which are not covered in the current paper.…”

mentioning

confidence: 99%

Fisher-KPP dynamics in diffusive Rosenzweig–MacArthur and Holling–Tanner models

Cai

Ghazaryan

Manukian

2019

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

We prove the existence of traveling fronts in diffusive Rosenzweig-MacArthur and Holling-Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig-MacArthur predatorprey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts with the fronts in a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. We obtain a similar result for a diffusive Holling-Tanner population model. In the second situation for the Rosenzweig-MacArthur model we prove the existence of the fronts but without observing a direct relation with Fisher-KPP equation. The analysis suggests that, in a variety of reaction-diffusion systems that rise in population modeling, parameter regimes may be found when the dynamics of the system is inherited from the scalar Fisher-KPP equation.Reaction-diffusion systems are often used in population dynamics modeling when it is desirable to take into account random motion of individuals in the population. In systems where there is more than one spatially homogeneous equilibrium state, it is of interest to know whether transition fronts between these states exist. We present analysis of traveling fronts in two diffusive populationdynamics models, one of which is a modified Rosenzweig-MacArthur model, and the other is Holling-Tanner predator-prey model. The analysis is performed in detail on the Rosenzweig-MacArthur system (Section 2), while for the Holling-Tanner system (Section 6) most of the details are skipped and similarities in the proofs are pointed out. The plan of the paper is as follows. We introduce the Rosenzweig-MacArthur model in Section 2.1 and explain the results of the paper and their mathematical implications. The scaling of the model that we use and the parameter regimes that the analysis covers are described in Section 2.2. The regimes are grouped in two cases that are then analyzed using geometric singular perturbation theory in Sections 3 and 4. In Section 3 the relation of the fronts with a scalar Fisher-KPP equation is revealed. In the analysis of the Rosenzweig-MacArthur model, assumptions are made about some of the parameters, in order to simplify some of the calculations. In Section 5, we describe the implications of the assumptions and Date

show abstract

Absolute stability and dynamical stabilisation in predator-prey systems

Cited by 18 publications

References 32 publications

A Mathematical Biologist’s Guide to Absolute and Convective Instability

A Mathematical Biologist’s Guide to Absolute and Convective Instability

Dynamical stabilization and traveling waves in integrodifference equations

Fisher-KPP dynamics in diffusive Rosenzweig–MacArthur and Holling–Tanner models

Contact Info

Product

Resources

About